Question

PROBLEM 2.6 A control rod made of yellow brass must not stretch more than ( 3 mathrm{~mm} ) when the tension in the wire is ( 4 mathrm{kN} ). Knowing that ( E=105 mathrm{GPa} ) and that the maximum allowable normal stress is ( 180 mathrm{MPa} ), determine (a) the smallest diameter rod that should be used, (b) the corresponding maximum length of the rod. SOLUTION (a) ( sigma=frac{P}{A} ; quad A=frac{pi d^{2}}{4} ) Substituting, we have [ egin{aligned} sigma=frac{P}{left(frac{pi d^{2}}{4} ight)} Rightarrow d & =sqrt{frac{4 P}{sigma pi}} \ d & =sqrt{frac{4left(4 imes 10^{3} mathrm{~N} ight)}{left(180 imes 10^{6} mathrm{~Pa} ight) pi}} \ d & =5.3192 imes 10^{-3} mathrm{~m} end{aligned} ] [ d=5.32 mathrm{~mm} ] (b) ( quad sigma=E varepsilon ; quad varepsilon=frac{delta}{L} ) Substituting, we have [ egin{aligned} sigma=E frac{delta}{L} Rightarrow L & =frac{E partial}{sigma} \ L & =frac{left(105 imes 10^{9} mathrm{~Pa} ight)left(3 imes 10^{-3} mathrm{~m} ight)}{left(180 imes 10^{6} mathrm{~Pa} ight)} quad L=1.750 mathrm{~m} end{aligned} ]

          PROBLEM 2.6
A control rod made of yellow brass must not stretch more than ( 3 mathrm{~mm} ) when the tension in the wire is ( 4 mathrm{kN} ). Knowing that ( E=105 mathrm{GPa} ) and that the maximum allowable normal stress is ( 180 mathrm{MPa} ), determine (a) the smallest diameter rod that should be used, (b) the corresponding maximum length of the rod.
SOLUTION
(a) ( sigma=frac{P}{A} ; quad A=frac{pi d^{2}}{4} )
Substituting, we have
[
egin{aligned}
sigma=frac{P}{left(frac{pi d^{2}}{4}
ight)} Rightarrow d & =sqrt{frac{4 P}{sigma pi}} \
d & =sqrt{frac{4left(4 	imes 10^{3} mathrm{~N}
ight)}{left(180 	imes 10^{6} mathrm{~Pa}
ight) pi}} \
d & =5.3192 	imes 10^{-3} mathrm{~m}
end{aligned}
]
[
d=5.32 mathrm{~mm}
]
(b) ( quad sigma=E varepsilon ; quad varepsilon=frac{delta}{L} )
Substituting, we have
[
egin{aligned}
sigma=E frac{delta}{L} Rightarrow L & =frac{E partial}{sigma} \
L & =frac{left(105 	imes 10^{9} mathrm{~Pa}
ight)left(3 	imes 10^{-3} mathrm{~m}
ight)}{left(180 	imes 10^{6} mathrm{~Pa}
ight)} quad L=1.750 mathrm{~m}
end{aligned}
]

$PROBLEM 2.6 A control rod made of yellow brass must not stretch more than ( 3 mathrm mm ) when the tension in the wire is ( 4 mathrmkN ). Knowing that ( E=105 mathrmGPa ) and that the maximum allowable normal stress is ( 180 mathrmMPa ), determine (a) the smallest diameter rod that should be used, (b) the corresponding maximum length of the rod. SOLUTION (a) ( sigma=fracPA ; quad A=fracpi d^24 ) Substituting, we have [ eginaligned sigma=fracPleft(fracpi d^24 ight) Rightarrow d =sqrtfrac4 Psigma pi d =sqrtfrac4left(4 imes 10^3 mathrm N ight)left(180 imes 10^6 mathrm Pa ight) pi d =5.3192 imes 10^-3 mathrm m endaligned ] [ d=5.32 mathrm mm ] (b) ( quad sigma=E varepsilon ; quad varepsilon=fracdeltaL ) Substituting, we have [ eginaligned sigma=E fracdeltaL Rightarrow L =fracE partialsigma L =fracleft(105 imes 10^9 mathrm Pa ight)left(3 imes 10^-3 mathrm m ight)left(180 imes 10^6 mathrm Pa ight) quad L=1.750 mathrm m endaligned ]$

Added by Ibrahim A.

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